The formula (2) shows that a system of concurrent forces represented by m1·OP>1, m2·OP>2, ... mn·OP>n will have a resultant represented hy Σ(m)·OG>. If we imagine O to recede to infinity in any direction we learn that a system of parallel forces proportional to m1, m2,... mn, acting at P1, P2 ... Pn have a resultant proportional to Σ(m) which acts always through a point G fixed relatively to the given mass-system. This contains the theory of the “centre of gravity” (§§ 4, 9). We may note also that if P1, P2, ... Pn, and P1′, P2′, ... Pn′ represent two configurations of the series of particles, then
Σ(m·PP>′) = Sigma(m)·GG>′,
(8)
where G, G′ are the two positions of the mass-centre. The forces m1·P>1P1′, m2·P>2P2′, ... mn·P>nPn′, considered as localized vectors, do not, however, as a rule reduce to a single resultant.
We proceed to the theory of the plane, axial and polar quadratic moments of the system. The axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions. If h1, h2, ... hn be the perpendicular distances of the particles from any fixed plane, the sum Σ(mh2) is the quadratic moment with respect to the plane. If p1, p2, ... pn be the perpendicular distances from any given axis, the sum Σ(mp2) is the quadratic moment with respect to the axis; it is also called the moment of inertia about the axis. If r1, r2, ... rn be the distances from a fixed point, the sum Σ(mr2) is the quadratic moment with respect to that point (or pole). If we divide any of the above quadratic moments by the total mass Σ(m), the result is called the mean square of the distances of the particles from the respective plane, axis or pole. In the case of an axial moment, the square root of the resulting mean square is called the radius of gyration of the system about the axis in question. If we take rectangular axes through any point O, the quadratic moments with respect to the co-ordinate planes are
Ix = Σ(mx2), Iy = Σ(my2), Iz = Σ(mz2);
(9)
those with respect to the co-ordinate axes are
Iyz = Σ {m (y2 + z2)}, Izx = Σ {m (z2 + x2)}, Ixy = Σ {m (x2 + y2)};
(10)